John Willard Milnor | |
---|---|

Born | |

Nationality | American |

Alma mater | Princeton University (AB, PhD) |

Known for | Exotic spheres Fary–Milnor theorem Hauptvermutung Milnor K-theory Microbundle Milnor Map Milnor's theorem ^{ [1] }Milnor–Thurston kneading theory Milnor–Wood inequality Surgery theory Švarc–Milnor lemma |

Spouse(s) | Dusa McDuff |

Awards | Putnam Fellow (1949, 1950) Sloan Fellowship (1955) Fields Medal (1962) National Medal of Science (1967) Leroy P. Steele Prize (1982, 2004, 2011) Wolf Prize (1989) Abel Prize (2011) |

Scientific career | |

Fields | Mathematics |

Institutions | Stony Brook University |

Doctoral advisor | Ralph Fox |

Doctoral students | Tadatoshi Akiba Jon Folkman John Mather Laurent C. Siebenmann Michael Spivak |

**John Willard Milnor** (born February 20, 1931) is an American mathematician known for his work in differential topology, K-theory and dynamical systems. Milnor is a distinguished professor at Stony Brook University and one of the six mathematicians to have won the Fields Medal, the Wolf Prize, and the Abel Prize.

Milnor was born on February 20, 1931 in Orange, New Jersey.^{ [2] } His father was J. Willard Milnor and his mother was Emily Cox Milnor.^{ [3] }^{ [4] } As an undergraduate at Princeton University he was named a Putnam Fellow in 1949 and 1950 and also proved the Fary–Milnor theorem. Milnor graduated with an A.B. in mathematics in 1951 after completing a senior thesis, titled "Link groups", under the supervision of Robert H. Fox.^{ [5] } He remained at Princeton to pursue graduate studies and received his Ph.D. in mathematics in 1954 after completing a doctoral dissertation, titled "Isotopy of links", also under the supervision of Fox.^{ [6] } His dissertation concerned link groups (a generalization of the classical knot group) and their associated link structure. Upon completing his doctorate, he went on to work at Princeton. He was a professor at the Institute for Advanced Study from 1970 to 1990.

His students have included Tadatoshi Akiba, Jon Folkman, John Mather, Laurent C. Siebenmann, and Michael Spivak. His wife, Dusa McDuff, is a professor of mathematics at Barnard College.

One of his published works is his proof in 1956 of the existence of 7-dimensional spheres with nonstandard differential structure. Later, with Michel Kervaire, he showed that the 7-sphere has 15 differentiable structures (28 if one considers orientation).

An *n*-sphere with nonstandard differential structure is called an exotic sphere, a term coined by Milnor. He gave a complete inventory of differentiable structures in spheres of all dimensions with Kervaire, and only continued till 2009.

Egbert Brieskorn found simple algebraic equations for 28 complex hypersurfaces in complex 5-space such that their intersection with a small sphere of dimension 9 around a singular point is diffeomorphic to these exotic spheres. Subsequently, Milnor worked on the topology of isolated singular points of complex hypersurfaces in general, developing the theory of the Milnor fibration whose fiber has the homotopy type of a bouquet of *μ* spheres where *μ* is known as the Milnor number. Milnor's 1968 book on his theory inspired the growth of a huge and rich research area that continues to mature to this day.

In 1961 Milnor disproved the Hauptvermutung by illustrating two simplicial complexes that are homeomorphic but combinatorially distinct.^{ [7] }^{ [8] }

Milnor introduced the growth invariant in a finitely presented group and the theorem stating that the fundamental group of a negatively curved Riemannian manifold has exponential growth became a striking point in the foundation for modern geometric group theory, and the basis for the theory of a hyperbolic group in 1987 by Mikhail Gromov.

In 1984 Milnor introduced a definition of attractor.^{ [9] } The objects generalize standard attractors, include so-called unstable attractors and are now known as Milnor attractors.

Milnor's current interest is dynamics, especially holomorphic dynamics. His work in dynamics is summarized by Peter Makienko in his review of *Topological Methods in Modern Mathematics*:

It is evident now that low-dimensional dynamics, to a large extent initiated by Milnor's work, is a fundamental part of general dynamical systems theory. Milnor cast his eye on dynamical systems theory in the mid-1970s. By that time the Smale program in dynamics had been completed. Milnor's approach was to start over from the very beginning, looking at the simplest nontrivial families of maps. The first choice, one-dimensional dynamics, became the subject of his joint paper with Thurston. Even the case of a unimodal map, that is, one with a single critical point, turns out to be extremely rich. This work may be compared with Poincaré's work on circle diffeomorphisms, which 100 years before had inaugurated the qualitative theory of dynamical systems. Milnor's work has opened several new directions in this field, and has given us many basic concepts, challenging problems and nice theorems.

^{ [10] }

His other significant contributions include microbundles, influencing the usage of Hopf algebras, algebraic K-theory, etc. He was an editor of the * Annals of Mathematics * for a number of years after 1962. He has written a number of books. He served as Vice President of the AMS in 1976–77 period.

Milnor was elected as a member of the American Academy of Arts and Sciences in 1961.^{ [11] } In 1962 Milnor was awarded the Fields Medal for his work in differential topology. He later went on to win the National Medal of Science (1967), the Lester R. Ford Award in 1970^{ [12] } and again in 1984,^{ [13] } the Leroy P. Steele Prize for "Seminal Contribution to Research" (1982), the Wolf Prize in Mathematics (1989), the Leroy P. Steele Prize for Mathematical Exposition (2004), and the Leroy P. Steele Prize for Lifetime Achievement (2011) "... for a paper of fundamental and lasting importance, On manifolds homeomorphic to the 7-sphere, Annals of Mathematics 64 (1956), 399–405".^{ [14] } In 1991 a symposium was held at Stony Brook University in celebration of his 60th birthday.^{ [15] }

Milnor was awarded the 2011 Abel Prize,^{ [16] } for his "pioneering discoveries in topology, geometry and algebra."^{ [17] } Reacting to the award, Milnor told the * New Scientist * "It feels very good," adding that "[o]ne is always surprised by a call at 6 o'clock in the morning."^{ [18] } In 2013 he became a fellow of the American Mathematical Society, for "contributions to differential topology, geometric topology, algebraic topology, algebra, and dynamical systems".^{ [19] }

- Milnor, John W. (1963).
*Morse theory*. Annals of Mathematics Studies, No. 51. Notes by M. Spivak and R. Wells. Princeton, NJ: Princeton University Press. ISBN 0-691-08008-9.^{ [20] } - —— (1965).
*Lectures on the h-cobordism theorem*. Notes by L. Siebenmann and J. Sondow. Princeton, NJ: Princeton University Press. ISBN 0-691-07996-X. OCLC 58324. - —— (1968).
*Singular points of complex hypersurfaces*. Annals of Mathematics Studies, No. 61. Princeton, NJ: Princeton University Press; Tokyo: University of Tokyo Press. ISBN 0-691-08065-8. - —— (1971).
*Introduction to algebraic K-theory*. Annals of Mathematics Studies, No. 72. Princeton, NJ: Princeton University Press. ISBN 978-0-691-08101-4. - Husemoller, Dale; Milnor, John W. (1973).
*Symmetric bilinear forms*. New York, NY: Springer-Verlag. ISBN 978-0-387-06009-5. - Milnor, John W.; Stasheff, James D. (1974).
*Characteristic classes*. Annals of Mathematics Studies, No. 76. Princeton, NJ: Princeton University Press; Tokyo: University of Tokyo Press. ISBN 0-691-08122-0.^{ [21] } - Milnor, John W. (1997) [1965].
*Topology from the differentiable viewpoint*. Princeton Landmarks in Mathematics. Princeton, NJ: Princeton University Press. ISBN 0-691-04833-9. - —— (1999).
*Dynamics in one complex variable*. Wiesbaden, Germany: Vieweg. ISBN 3-528-13130-6.*2nd edn*. 2000.^{ [22] }

- Milnor, John W. (1956). "On manifolds homeomorphic to the 7-sphere".
*Annals of Mathematics*. Princeton University Press.**64**(2): 399–405. doi:10.2307/1969983. JSTOR 1969983. MR 0082103. - —— (1959). "Sommes de variétés différentiables et structures différentiables des sphères".
*Bulletin de la Société Mathématique de France*. Société Mathématique de France.**87**: 439–444. doi: 10.24033/bsmf.1538 . MR 0117744. - —— (1959b). "Differentiable structures on spheres".
*American Journal of Mathematics*. Johns Hopkins University Press.**81**(4): 962–972. doi:10.2307/2372998. JSTOR 2372998. MR 0110107. - —— (1961). "Two complexes which are homeomorphic but combinatorially distinct".
*Annals of Mathematics*. Princeton University Press.**74**(2): 575–590. doi:10.2307/1970299. JSTOR 1970299. MR 0133127. - —— (1984). "On the concept of attractor".
*Communications in Mathematical Physics*. Springer Press.**99**(2): 177–195. Bibcode:1985CMaPh..99..177M. doi:10.1007/BF01212280. MR 0790735. - Kervaire, Michel A.; Milnor, John W. (1963). "Groups of homotopy spheres: I" (PDF).
*Annals of Mathematics*. Princeton University Press.**77**(3): 504–537. doi:10.2307/1970128. JSTOR 1970128. MR 0148075.CS1 maint: ref=harv (link) - Milnor, John W. (2011). "Differential topology forty-six years later" (PDF).
*Notices of the American Mathematical Society*.**58**(6): 804–809.

- Milnor, John Willard; Munkres, James Raymond (2007). "Lectures on Differential Topology". In Milnor, John Willard (ed.).
*Collected papers of John Milnor, Volume 4*. American Mathematical Society. pp. 145–176. ISBN 978-0-8218-4230-0.

In mathematics, **differential topology** is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

In mathematics, **topology** is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.

**Pierre René, Viscount Deligne** is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal.

**Norman Earl Steenrod** was an American mathematician most widely known for his contributions to the field of algebraic topology.

In mathematics, **geometric topology** is the study of manifolds and maps between them, particularly embeddings of one manifold into another.

**Curtis Tracy McMullen** is an American mathematician who is the Cabot Professor of Mathematics at Harvard University. He was awarded the Fields Medal in 1998 for his work in complex dynamics, hyperbolic geometry and Teichmüller theory.

In differential topology, an **exotic sphere** is a differentiable manifold *M* that is homeomorphic but not diffeomorphic to the standard Euclidean *n*-sphere. That is, *M* is a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar one.

In mathematics, topology generalizes the notion of triangulation in a natural way as follows:

In mathematics, a **manifold** is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or *n-manifold* for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to the Euclidean space of dimension n.

The **Hauptvermutung** of geometric topology is the question of whether any two triangulations of a triangulable space have subdivisions that are combinatorially equivalent, i.e. the subdivided triangulations are built up in the same combinatorial pattern.

In mathematics, a differentiable manifold of dimension *n* is called **parallelizable** if there exist smooth vector fields

**Phillip Augustus Griffiths IV** is an American mathematician, known for his work in the field of geometry, and in particular for the complex manifold approach to algebraic geometry. He was a major developer in particular of the theory of variation of Hodge structure in Hodge theory and moduli theory. He also worked on partial differential equations, coauthored with Shiing-Shen Chern, Robert Bryant and Robert Gardner on Exterior Differential Systems.

**William Browder** is an American mathematician, specializing in algebraic topology, differential topology and differential geometry. Browder was one of the pioneers with Sergei Novikov, Dennis Sullivan and C. T. C. Wall of the surgery theory method for classifying high-dimensional manifolds. He served as President of the American Mathematical Society until 1990.

**James Raymond Munkres** is a Professor Emeritus of mathematics at MIT and the author of several texts in the area of topology, including *Topology*, *Analysis on Manifolds*, *Elements of Algebraic Topology*, and *Elementary Differential Topology*. He is also the author of *Elementary Linear Algebra*.

In the mathematical area of topology, the **generalized Poincaré conjecture** is a statement that a manifold which is a homotopy sphere *is* a sphere. More precisely, one fixes a category of manifolds: topological (**Top**), piecewise linear (**PL**), or differentiable (**Diff**). Then the statement is

**John Coleman Moore** was an American mathematician. The Borel−Moore homology and Eilenberg–Moore spectral sequence are named after him.

**Michel André Kervaire** was a French mathematician who made significant contributions to topology and algebra.

In mathematics, the **Kervaire invariant** is an invariant of a framed -dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. This invariant was named after Michel Kervaire who built on work of Cahit Arf.

**Barry Charles Mazur** is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in number theory, Mazur's torsion theorem in arithmetic geometry, the Mazur swindle in geometric topology, and the Mazur manifold in differential topology.

**James Dillon Stasheff** is an American mathematician, a professor emeritus of mathematics at the University of North Carolina at Chapel Hill. He works in algebraic topology and algebra as well as their applications to physics.

- ↑ Milnor's Theorem – from Wolfram MathWorld
- ↑ Staff.
*A COMMUNITY OF SCHOLARS: The Institute for Advanced Study Faculty and Members 1930–1980*, p. 35. Institute for Advanced Study, 1980. Accessed November 24, 2015. "Milnor, John Willard M, Topology Born 1931 Orange, NJ." - ↑ Helge Holden; Ragni Piene (3 February 2014).
*The Abel Prize 2008–2012*. Springer Berlin Heidelberg. pp. 353–360. ISBN 978-3-642-39448-5. - ↑ Allen G. Debus (1968).
*World Who's who in Science: A Biographical Dictionary of Notable Scientists from Antiquity to the Present*. Marquis-Who's Who. p. 1187. - ↑ Milnor, John W. (1951).
*Link groups*. Princeton, NJ: Department of Mathematics. - ↑ Milnor, John W. (1954).
*Isotopy of links*. Princeton, NJ: Department of Mathematics. - ↑ : –.Cite journal requires
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(help) - ↑ Milnor, John (1985). "On the concept of attractor".
*Communications in Mathematical Physics*.**99**(2): 177–195. Bibcode:1985CMaPh..99..177M. doi:10.1007/BF01212280. ISSN 0010-3616. - ↑ Lyubich, Mikhail (1993). Michael Yampolsky (ed.).
*Holomorphic Dynamics and Renormalization: A Volume in Honour of John Milnor's 75th Birthday*. Houston, Texas. pp. 85–92. - ↑ "John Willard Milnor".
*American Academy of Arts & Sciences*. Retrieved 2020-05-31. - ↑ Milnor, John (1969). "A problem in cartography".
*Amer. Math. Monthly*.**76**(10): 1101–1112. doi:10.2307/2317182. JSTOR 2317182. - ↑ Milnor, John (1983). "On the geometry of the Kepler problem".
*Amer. Math. Monthly*.**90**(6): 353–365. doi:10.2307/2975570. JSTOR 2975570. - ↑ O'Connor, J J; EF Robertson. "John Willard Milnor".
- ↑ Goldberg, Lisa R.; Phillips, Anthony V., eds. (1993),
*Topological methods in modern mathematics*, Proceedings of the symposium in honor of John Milnor's sixtieth birthday held at the State University of New York, Stony Brook, New York, June 14–21, 1991, Houston, TX: Publish-or-Perish Press, ISBN 978-0-914098-26-3 - ↑ Abelprisen (Abel Prize) website. "The Abel Prize awarded to John Milnor, Stony Brook University, NY". Archived from the original on April 29, 2011. Retrieved March 24, 2011.
- ↑ Ramachandran, R. (March 24, 2011). "Abel Prize awarded to John Willard Milnor".
*The Hindu*. Retrieved 24 March 2011. - ↑ Aron, Jacob (March 23, 2011). "Exotic sphere discoverer wins mathematical 'Nobel'".
*New Scientist*. Retrieved 24 March 2011. - ↑ 2014 Class of the Fellows of the AMS, American Mathematical Society, retrieved 2013-11-04.
- ↑ Kuiper, N. H. (1965). "Review:
*Morse theory*, by John Milnor".*Bull. Amer. Math. Soc*.**71**(1): 136–137. doi: 10.1090/s0002-9904-1965-11251-4 . - ↑ Spanier, E. H. (1975). "Review:
*Characteristic classes*, by John Milnor and James D. Stasheff".*Bull. Amer. Math. Soc*.**81**(5): 862–866. doi: 10.1090/s0002-9904-1975-13864-x . - ↑ Hubbard, John (2001). "Review:
*Dynamics in one complex variable*, by John Milnor".*Bull. Amer. Math. Soc. (N.S.)*.**38**(4): 495–498. doi: 10.1090/s0273-0979-01-00918-1 .

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- O'Connor, John J.; Robertson, Edmund F., "John Milnor",
*MacTutor History of Mathematics archive*, University of St Andrews . - Home page at SUNYSB
- Photo
- Exotic spheres home page
- The Abel Prize 2011 – video
- Raussen, Martin; Skau, Christian (March 2012). "Interview with John Milnor" (PDF).
*Notices of the American Mathematical Society*.**59**(3): 400–408. doi:10.1090/noti803.

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